Normal form and existence time for the Kirchhoff equation

CDSNS Colloquium
Friday, April 23, 2021 - 1:00pm for 1 hour (actually 50 minutes)
Zoom (see additional notes for link)
Emanuele Haus – University of Roma Tre – ehaus@mat.uniroma3.it
Alex Blumenthal

Please Note: Zoom link:

In this talk I will present some recent results on the Kirchhoff equation with periodic boundary conditions, in collaboration with Pietro Baldi.

Computing the first step of quasilinear normal form, we erase from the equation all the cubic terms giving nonzero contribution to the energy estimates; thus we deduce that, for small initial data of size $\varepsilon$ in Sobolev class, the time of existence of the solution is at least of order $\varepsilon^{-4}$ (which improves the lower bound $\varepsilon^{-2}$ coming from the linear theory).

In the second step of normal form, there remain some resonant terms (which cannot be erased) that give a non-trivial contribution to the energy estimates; this could be interpreted as a sign of non-integrability of the equation. Nonetheless, we show that small initial data satisfying a suitable nonresonance condition produce solutions that exist over a time of order at least $\varepsilon^{-6}$.